Nondictatorial & Nonmanipulable axioms (was Re: York's

[Bruce Anderson]

On May 4,  1:53am, Steve Eppley wrote:
> Subject: Re: Nondictatorial & Nonmanipulable axioms (was Re: York's
> Bruce Anderson wrote:
> [snip]
> >according to every definition I have seen here (or elsewhere),
> >Condorcet's voting method does not fail the non-dictatorial axiom;
> >instead, Condorcet's voting method fails the non-manipulable axiom. 
> [snip]
> 
> Are you talking Arrow, Gibbard-Satterthwaite, or both?

Gibbard-Satterthwaite.

>  Which of Arrow's 5 axioms does Condorcet violate?
 
Different authors structure Arrow's axioms different ways, so it can be 
important to specify a particular structure (or alternative set of structures) 
when discussing these axioms.  In terms of Ordeshook's structure (which you 
wisely provide below), Condorcet's method violates A4 (Independence).  This 
axiom is frequently called the Independence of Irrelevant Alternatives axiom (or 
criterion).

> Another Saari (Mike, in the eVote maillist) claims that, by Arrow's
> theorem, ranked ballot methods like Condorcet must be "dictatorial".
> I'd certainly appreciate learning the truth.

What is the eVote maillist?

> My understanding is that axiom A4 (below) limits the voting method
> to using only ranking info from the voters the way Condorcet does,
> ignoring any rating (a.k.a. weighting, a.k.a. intensity) info.

No, it's A2 that does this.
  
> And that the purpose of this limitation is to prevent voters from
> manipulating the outcome by strategically misrepresenting their
> ratings.  

That's Gibbard-Satterthwaite, not Arrow.

> Bruce, are you saying that Condorcet violates A4 and not A5? 

You got it!  I think (is the e-mailese "imho" correct here?), you were 
hitting on a key point when you said "... the Nondictatorial axiom is another 
one of these academic yes/no criteria:  Is there a possible scenario where..."  
Many (if not most) of these "impossibile-to-satisfy" types of theorems consist 
of several axioms that are either quite reasonable, or needed just to provide 
structure, or both, and one axiom that is of the "Is there a possible scenario 
where ..." type.  Typically, all the "even-remotely-reasonable" voting methods, 
like Condorcet and Copeland, fail the "Is there a possible scenario where ..." 
axiom.  For Gibbard-Satterthwaite, that axiom is the non-manipulable axiom ; for 
Arrow, it's A4 below.

> - -
> 
> Arrow's 5 axioms [according to Ordershook, P.C., "Game Theory and Political   
  Theory," Cambridge, 1986]:
>
> A1. (Collective rationality)  The social preference function 
> satisfies the following two assumptions:
> 
>    (Completeness)  For every pair of outcomes o1 and o2,
>                    either o1 is liked at least as much as o2, 
>                    or o2 is liked at least as much as o1.  
> 
>    (Transitivity)  For any three outcomes o1, o2, and o3, 
>                    if o1 is liked at least as much as o2,
>                    and o2 is liked at least as much as o3, 
>                    then o1 is liked at least as much as o3.  
> 
> A2. (Unrestricted domain)  Every individual preference relation that 
> satisfies the two assumptions in A1 is admissible.
> 
> A3. (Pareto principle)  If every person in the group likes
> alternative x more than y, then x is preferred to y in the social
> preference order.  
> 
> A4. (Independence)  If R is a profile of individual preferences 
> over some set of alternatives that includes x and y, 
> if G(R,{x,y}) = (x is liked more than y), 
> and if R' is another preference profile such that each person's 
> preference between x and y is the same in R' as in R, 
> then G(R',{x,y}) = (x is liked more than y).  
> 
> Note: G(R,O) --> O is a social choice function that takes individual
> preferences and selects outcomes in O as the social outcome.
> 
> A5. (Nondictatorship)  No one person is decisive for every pair of 
> outcomes.
> (I.e., no one can get his/her way if opposed by everyone else.)
> 
> Arrow's Impossibility Theorem: If there are at least three possible
> outcomes, then voting methods which satisfy A1 through A4 violate A5.
> 
> --Steve
> 
>-- End of excerpt from Steve Eppley

Bruce